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This is one of those results in calculus which were anticipated by
Archimedes. He gave a correct formula for the volume but it is not known exactly
how Archimedes solved this problem. There is, however, a simple way to
obtain the answer without much calculus. Let me quote from late Gardner's [*The Unexpected Hangin... |
Let M, K, and L be isomorphic normal subgroups of a group G. Suppose $M \cap K \leq M \cap L$. Can we conclude anything about the index of MK in G and the index of ML in G? I would like it to be the case that MK has smaller index than ML, but I'm having a hard time convincing myself either way.
(EDIT: I initially le... |
This is one of those results in calculus which were anticipated by
Archimedes. He gave a correct formula for the volume but it is not known exactly
how Archimedes solved this problem. There is, however, a simple way to
obtain the answer without much calculus. Let me quote from late Gardner's [*The Unexpected Hangin... |
How can we show that every automorphism of S4 is an inner automorphism ? |
Inspired by [this question][1].
Let's consider the 11-letter word "MATHEMATICS". The general question is to find how many words it is possible to have if we can make 4 permutations of letters.
Sub-question a)
Each letter has a position in the word, thus when I say 1 permutation, we exchange the position of 2 l... |
Permutation of 4 letters in a 11-letter word : how many different words can we make? |
Let $\mathbf{x} =$ \{$x_{i}$} be a set of $n$ positive reals. In every good book on inequalities, one finds the classical result
\begin{eqnarray}
AM(\mathbf{x}) \geq GM(\mathbf{x}) \geq HM(\mathbf{x}),
\end{eqnarray}
where $AM(\mathbf{x}) = \frac{1}{n} \sum_{i = 1}^{n} x_{i}$ is the arithmetic mean, $GM(\mathbf{x}... |
Assume Q is a convex central symmetric curve, whose area is S.
The area of the maximum parallelogram inside Q is S'.
How to prove the conjecture that S'/S >= 2/pai=0.6366?
For example, If Q is an ellipse, S'=2ab, S=pai*ab.
If Q is a regular hexagon, S'/S=2/3.
It's trivial that S'/S>=1/2, and I know how to p... |
Assume Q is a convex central symmetric curve, whose area is $\displaystyle S$.
The area of the maximum parallelogram inside Q is $\displaystyle S'$.
How to prove the conjecture that $\displaystyle \frac{S'}{S} \ge \frac{2}{\pi}=0.6366\dots$?
For example, If Q is an ellipse, $\displaystyle S'=2ab$, $\displaystyle... |
Let $X$ denote a metric space. One can assume that $X$ is Polish if that helps, but I was trying to avoid to assume that $X$ is compact. Let $P(X)$ denote the set of Borel probability measures on $X$. The weak*-topology on $P(X)$ is defined as usual: a net $(p_{\alpha})$ in $P(X)$ converges to $p \in P(X)$ iff $|\int f... |
How to show that $e^{x} \geq \left (1 + \frac{x}{n} \right) ^{n}$ holds for each non-negative real $x$ and each integer $n \geq 1$ ? I tried series and induction but got stuck. Can you please help? |
Wikipedia articles on "unit sphere" and "unit circle" say the radius is 1. Articles on the "unit square" and "unit cube" say the length of the side is 1. Would you expect a unit torus to have major *radius* 1 or major *diameter* 1?
Admittedly, a torus is a different animal than a sphere, but... It feels most natural... |
what size is a "unit torus"? |
Let $\mathbf{x} =$ \{$x_{i}$} be a set of $n$ positive reals. In every good book on inequalities, one finds the classical result
\begin{eqnarray}
AM(\mathbf{x}) \geq GM(\mathbf{x}) \geq HM(\mathbf{x}),
\end{eqnarray}
where $AM(\mathbf{x}) = \frac{1}{n} \sum_{i = 1}^{n} x_{i}$ is the arithmetic mean, $GM(\mathbf{x}... |
Wikipedia articles on "unit sphere" and "unit circle" say the radius is 1. Articles on the "unit square" and "unit cube" say the length of the side is 1. Would you expect a unit torus to have major *radius* 1 or major *diameter* 1?
Admittedly, a torus is a different animal than a sphere, but... It feels most natural... |
Assume Q is a convex central symmetric curve, whose area is $\displaystyle S$.
The area of the maximum parallelogram inside Q is $\displaystyle S'$.
How to prove the conjecture that $\displaystyle \frac{S'}{S} \ge \frac{2}{\pi}=0.6366\dots$?
For example, If Q is an ellipse, $\displaystyle S'=2ab$, $\displaystyle... |
How can I prove that there is no closed form solution to the equation $2^x + 3^x = 10$? |
Let X be a measure space and let $f_{n}$ be a sequence of functions which converge pointwise to a function f in $L^{p}(X)$ where $p>1$ and suppose $g_{n}$ is a sequence of functions which converge pointwise to a function $g$ in $L^{\frac{p}{p-1}}(X)$. Prove that:
$\displaystyle \lim_{n \to \infty} \int_{X} f_{n}(x) ... |
I am currently engaged in independent study of algebraic geometry, using Dan Bump's book. One of the exercises in it outlines a proof of the Krull Intersection Theorem, which [here] is the following:
> Let A be a Noetherian local ring with maximal ideal m, and let M be the intersection of all of the m^n. Then M = 0.... |
Why is the Artin-Rees lemma used here? |
I am currently engaged in independent study of algebraic geometry, using Dan Bump's book. One of the exercises in it outlines a proof of the Krull Intersection Theorem, which [here] is the following:
> Let A be a Noetherian local ring with maximal ideal m, and let M be the intersection of all of the m^n. Then M = 0.... |
I am currently engaged in independent study of algebraic geometry, using Dan Bump's book. One of the exercises in it outlines a proof of the Krull Intersection Theorem, which [here] is the following:
> Let $A$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$, and let $M$ be the intersection of all of the... |
Assume Q is a convex central symmetric curve, whose area is $\displaystyle S$.
The area of the maximum parallelogram inside Q is $\displaystyle S'$.
How to prove the conjecture that $\displaystyle \frac{S'}{S} \ge \frac{2}{\pi}=0.6366\dots$?
For example, If Q is an ellipse, $\displaystyle S'=2ab$, $\displaystyle... |
Give me a second to fix the tag I can never get it right the first time before commenting please!
Edit : Could someone fix the exponent? I'm not sure how to get it to display properly, either that or I will fix it later I am in a rush at the moment.
I am fairly certain that this converges. Using the ratio or ro... |
I want to prove that the series
$$ \sum_{n=1}^{\infty}(-1)^n\left(2^{1/n}-1\right)$$
converges.
Give me a second to fix the tag I can never get it right the first time before commenting please!
Edit : Could someone fix the exponent? I'm not sure how to get it to display properly, either that or I will fix it late... |
I am confused on how to apply ito's formula on certain problems, especially where expectations are involved. For example, if $W_t$ is a Weiner process and $X_t$ satisfies a below SDE:
$ dX_t = (X_t-\mu)dt + \sigma\sqrt{X_t}dW_t,~~~~~~ X_0 = x_o$
How do I find $\partial_t \phi$ or $\partial_\xi \phi$ where $\ph... |
Suppose $X$ and $Y$ are independent random variables. The mgf for $X$ is $M_{X}(t) = (1-t)^{-1}, \ t<1$. The mgf for $Y$ is $M_{Y}(t) = (1-2t)^{-1}, \ t< 0.5$. Two other random variables $U$ and $V$ are defined by $U = \frac{1}{2}(X+Y)$ and $V = \frac{1}{2}(X-Y)$. Calculate $\text{Cov}(U,V)$.
So $\text{Cov}(U,V) =... |
I am confused on how to apply ito's formula on certain problems, especially where expectations are involved. For example, if $W_t$ is a Weiner process and $X_t$ satisfies a below SDE:
$ dX_t = (X_t-\mu)dt + \sigma\sqrt{X_t}dW_t,~~~~~~ X_0 = x_o$
How do I find $\partial_t \phi$ or $\partial_\xi \phi$ where $\ph... |
I am confused on how to apply Ito's formula on certain problems, especially where expectations are involved. For example, if $W_t$ is a Weiner process and $X_t$ satisfies a below SDE:
$ dX_t = (X_t-\mu)dt + \sigma\sqrt{X_t}dW_t,~~~~~~ X_0 = x_o$
How do I find $\partial_t \phi$ or $\partial_\xi \phi$ where $\ph... |
Wikipedia articles on "unit sphere" and "unit circle" say the radius is 1. Articles on the "unit square" and "unit cube" say the length of the side is 1. Would you expect a unit torus to have major *radius* 1 or major *diameter* 1?
Admittedly, a torus is a different animal than a sphere, but... It feels most natural... |
Wikipedia articles on "unit sphere" and "unit circle" say the radius is 1. Articles on the "unit square" and "unit cube" say the length of the side is 1. Would you expect a unit torus to have major *radius* 1 or major *diameter* 1?
Admittedly, a torus is a different animal than a sphere, but... It feels most natural... |
Wikipedia articles on "unit sphere" and "unit circle" say the radius is 1. Articles on the "unit square" and "unit cube" say the length of the side is 1. Would you expect a unit torus to have major *radius* 1 or major *diameter* 1?
Admittedly, a torus is a different animal than a sphere, but... It feels most natural... |
I have a set of quadratic equations of the form..
$ 2S_0q_0(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_0 $
$ 2S_1q_1(q_*S_0 + q_1S_1 + ...... + q_iS_n)+k_1 $
.
.
.
$ 2S_nq_n(q_0S_0 + q_1S_1 + ...... + q_nS_n)+k_n $
We have n equations and n variables $q_0,q_1,q_2.....q_n$. The k and S values a... |
I have been thinking about the differentials that we use in derivatives and integrals.
For example, I have an equation:
${\int{w}{dr}}$ = other stuff
But w is ${\frac{dr}{dt}}$.
So if I can replace w with ${\frac{dr}{dt}}$, and differentials can be multiplied like normal, I would get ${\frac{dr^2}{dt}}$.
... |
To what extent can you manipulate differentials like dy and dt like actual values? |
You want to show that two things happen:
1. Every element of $V$ can be written as a sum $u+w$, where $u\in\mathrm{ker}(f-\mathrm{id}_V)$ and $v\in\mathrm{ker}(f+\ker{id}_v)$; and
2. The intersection of $\mathrm{ker}(f-\mathrm{id}_V)$ and $\mathrm{ker}(f+\mathrm{id}_V)$ is trivial (contains only the zero vector... |
I have a set of quadratic equations of the form..
$ 2S_0q_0(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_0 = 0$
$ 2S_1q_1(q_*S_0 + q_1S_1 + ...... + q_iS_n)+k_1 = 0$
.
.
.
$ 2S_nq_n(q_0S_0 + q_1S_1 + ...... + q_nS_n)+k_n = 0$
We have n equations and n variables $q_0,q_1,q_2.....q_n$. The k and S... |
I have a set of quadratic equations of the form..
$ 2S_0q_0(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_0 = 0$
$ 2S_1q_1(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_1 = 0$
.
.
.
$ 2S_nq_n(q_0S_0 + q_1S_1 + ...... + q_nS_n)+k_n = 0$
We have n equations and n variables $q_0,q_1,q_2.....q_n$. The k and S... |
You want to show that two things happen:
1. Every element of $V$ can be written as a sum $u+w$, where $u\in\mathrm{ker}(f-\mathrm{id}_V)$ and $v\in\mathrm{ker}(f+\ker{id}_v)$; and
2. The intersection of $\mathrm{ker}(f-\mathrm{id}_V)$ and $\mathrm{ker}(f+\mathrm{id}_V)$ is trivial (contains only the zero vect... |
Let $(X, \prec)$ and $(Y, <)$ be two well-ordered sets. Now, I want to understand the proof that $X$ is isomorphic to an initial segment of $Y$ or $Y$ is isomorphic with an initial segment of $X$.
Now, they define $\mathcal{F}$ to be the set of all mappings with as domain an initial segment of $X$ and $Y$ as codomai... |
It comes down to whether or not you want $K(X)$ to be a group. If all you care about is isomorphism classes of vector bundles, that's a perfectly good monoid. People wouldn't call the monoid $K$-theory because $K$-theory of stuff generally means you've performed some kind of group-completion of a monoid or a category... |
Let $(X, \prec)$ and $(Y, <)$ be two well-ordered sets. Now, I want to understand the proof that $X$ is isomorphic to an initial segment of $Y$ or $Y$ is isomorphic with an initial segment of $X$.
Now, they define $\mathcal{F}$ to be the set of all mappings with as domain an initial segment of $X$ and $Y$ as codomai... |
If I have two discrete distributions of random variable X with prob. mass functions u(X) and v(X), such that,
v($X=X_i$) = $\lambda\frac{u(X=X_i)}{X_i} $
what can I infer from this equation?
What if,
v($X=X_i$) = $\lambda\sqrt{\frac{u(X=X_i)}{X_i} }$
In both cases, $\lambda$ is a constant. |
Let $(X, \prec)$ and $(Y, <)$ be two well-ordered sets. Now, I want to understand the proof that $X$ is isomorphic to an initial segment of $Y$ or $Y$ is isomorphic with an initial segment of $X$.
Now, they define $\mathcal{F}$ to be the set of all mappings with as domain an initial segment of $X$ and $Y$ as codomai... |
I hope this question is not too general, but I am not completely sure yet how to phrase it.
So we all know that when we have two sequences of random variables $X_n$ and $Y_n$ for $n \ge 1$, that converge to $X$ and $Y$ (using some measure, almost surely, or with probability, in distribution, etc.) then we have all k... |
convergence of random variables? |
It comes down to whether or not you want $K(X)$ to be a group. If all you care about is isomorphism classes of vector bundles, that's a perfectly good monoid. People wouldn't call the monoid $K$-theory because $K$-theory of stuff generally means you've performed some kind of group-completion of a monoid or a category... |
I don't have enough reputation yet, or this would be a comment.
A random variable can have only one probability mass function, so it is not clear what you are asking. Where do these equations come from? |
If I have two different discrete distributions of random variable X with prob. mass functions u(X) and v(X) (*from two different systems*), such that,
v($X=X_i$) = $\lambda\frac{u(X=X_i)}{X_i} $
what can I infer from this equation?
What if,
v($X=X_i$) = $\lambda\sqrt{\frac{u(X=X_i)}{X_i} }$
... |
How can I calculate the arc between two circles? The arc must be tangent to the two points on the circles.
Here is a picture illustrating it. I'm trying to code and calculate the orange arc and the blue arc.
https://i.stack.imgur.com/8GsDV.png
The points B, F, D, and E can be changed by the user and dynamical... |
How can I calculate the arc between two circles? The arc must be tangent to the two points on the circles.
Here is a picture illustrating it. I'm trying to code and calculate the orange arc and the blue arc.
![circle arcs][1]
The points B, F, D, and E can be changed by the user and dynamically update.
[1... |
Suppose that you have an e.g.f.: $E(z)=\sum_{n=0}^{\infty} \frac{a_{n}z^{n}}{n!}$, and that the definition of $a_{n}$ can be reasonably extended to noninteger arguments. (the Catalan numbers $C_{n}$, would be written in terms of the Gamma function thusly: $C_{n} = \frac{\Gamma(2n+1)}{\Gamma(n+2)\Gamma(n+1)}$, for insta... |
I'm a bit curious as to how "real" mathematicians would solve this problem.
"Find the least prime number greater than 2000."
Of course, I can always go brute force:
IF N IS PRIME, OUTPUT N
ELSE INCREASE N BY 1
But that's no fun. (Especially when the number is really high.)
Are there any algorith... |
Can someone show mathematically how gimbal lock happens when doing matrix rotation with euler angles for yaw, pitch, roll? I'm having a hard time understanding what is going on even after reading several articles on Google.
Is the only work-around to use quaternions? |
It comes down to whether or not you want $K(X)$ to be a group. If all you care about is isomorphism classes of vector bundles, that's a perfectly good monoid. People wouldn't call the monoid $K$-theory because $K$-theory of stuff generally means you've performed some kind of group-completion of a monoid or a category... |
"Euler Angles" you can think of as a function $(S^1)^3 \to SO_3$ or $\mathbb R^3 \to SO_3$. The derivative of this function does not always have rank 3, so you have degenerate submanifolds where the function is many-to-one. In this special case that's called "gimbal lock".
One formalism that avoids this is quat... |
"Euler Angles" you can think of as a function $(S^1)^3 \to SO_3$ or $\mathbb R^3 \to SO_3$. The derivative of this function does not always have rank 3, so you have degenerate submanifolds where the function is many-to-one. In this special case that's called "gimbal lock".
One formalism that avoids this is quat... |
If I have two different discrete distributions of random variables X and Y, such that their probability mass functions are related as follows:
$P(X=x_i) = \lambda\frac{P (Y=x_i)}{x_i} $
what can I infer from this equation? Any observations or interesting properties that you see based on this relation?
... |
AB+CD+A’BD+A’BC+AB’D+AB’C
AB+CD+A’B(D+C)+AB’(D+C)
CD+(AB+A'B(D+C)+AB’(D+C))
CD+(AB+B(D+C)+A(D+C))
CD+AB+(B(D+C)+A(D+C))
CD+AB+(A+B)(D+C)
This is what I got, but I'm not sure if it's correct. It's as good as I can get it algebraically. Can it be simplified further? |
Boolean Algebra: Can this be simplified further? |
A'B'C'+AB'C'+ABC'
C'(A'B'+AB'+AB)
C'(A'B'+A(B'+B))
C'(A'B'+A)
C'(B'+A)
It's that last step that used to trip me up. `A'+AB = A'+B` Forget what that law is called (identity?). |
"Euler Angles" you can think of as a function $(S^1)^3 \to SO_3$ or $\mathbb R^3 \to SO_3$. The derivative of this function does not always have rank 3, so you have degenerate submanifolds where the function is many-to-one. In this special case that's called "gimbal lock".
One formalism that avoids this is quat... |
I'm trying to compute the following limit:
$\displaystyle \lim_{n \to \infty} \int_{0}^{\infty} (1+\frac{x}{n})^{(-n)} sin(x/n) dx$
Clearly the integrand converges to 0 pointwise so the trick is to find the dominating function. So for simplicity let ${f_{n}}$ denote the sequence of functions of the integrand: i... |
"Euler Angles" you can think of as a function $(S^1)^3 \to SO_3$ or $\mathbb R^3 \to SO_3$. The derivative of this function does not always have rank 3, so you have degenerate submanifolds where the function is many-to-one. In this special case that's called "gimbal lock".
One formalism that avoids this is quat... |
The Poincaré conjecture is listed as one of the Millennium Prize Problems and has received significant attention from the media a few years ago when Grigori Perelman presented a proof of this conjecture. But why is this interesting at all? What practical and interesting questions can be answered now that this conjectur... |
What is the importance of the Poincaré conjecture? |
"Euler Angles" you can think of as a function $(S^1)^3 \to SO_3$ or $\mathbb R^3 \to SO_3$. The derivative of this function does not always have rank 3, so you have degenerate submanifolds where the function is many-to-one. In this special case that's called "gimbal lock".
One formalism that avoids this is quat... |
This is an exercise from Rotman, An Introduction to Homological Algebra, which I've been thinking now and then for a few days and I haven't solved it yet. I've decided to ask here because it is bugging me and I don't have any friends also studying this subject (in case it is of any use, I'm an undergraduate).
Let $R... |
Suppose $G_1$ and $G_2$ are Lie groups with isomorphic Lie algebras. Then from standard Lie theory we know that there is a simply connected Lie group $G$ such that $G/H_i = G_i$ where $H_i$ is a discrete subgroup of the center of $G$. I am curious if there is a nice condition on $H_1,H_2$ and how they sit in $G$ that... |
When is $G/H = G/K$? In particular, when are two Lie groups with isomorphic Lie algebras isomorphic? |
I was experimenting with binary operators, and devised a commutative operator with function $f$ such that the operator mapped $f \colon \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{R}$ in order to make up some problem (e.g. if $f(3,10)=30$ and $f(4,3)=10$, find $f(4,6)$, etc.).
My question is, are there any func... |
First, you don't have the freedom to choose $\lambda$. (Maybe you already realize that?) In order for $P(X = x_i)$ to be a true probability distribution, you must have
$$\lambda = \left(\sum_{x_i} \frac{P(Y = x_i)}{x_i}\right)^{-1}.$$
Given that, if $Y$ is [geometric][1] with success probability $p$, and $P(X = ... |
This is an exercise from Rotman, An Introduction to Homological Algebra, which I've been thinking now and then for a few days and I haven't solved it yet. I've decided to ask here because it is bugging me and I don't have any friends also studying this subject (in case it is of any use, I'm an undergraduate).
Let $R... |
The *[harmonic mean][1]* of a finite set of positive real numbers $\\{x_1, x_2, \ldots, x_n\\}$ is defined to be $$H(\\{x_1, x_2, \ldots, x_n\\}) = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}}.$$
The *[logarithmic mean][2]* of two distinct positive real numbers $a$ and $b$ is defined to be $$L(a,... |
I hear people use these words relatively interchangeably. I'd believe that any differentiable manifold can also be made into a variety (which data, if I understand correctly, implicitly includes an ambient space?), but it's unclear to me whether the only non-varietable manifolds should be those that don't admit smooth... |
What is the difference between a variety and a manifold? |
I've been wondering if/when it's possible to "truncate" a series.
>Example 1
For example, the closed form for the series of naturals is:
$\frac{1}{(x-1)^2}$ = $1 + 2z + 3x^2 + \cdots$
The "truncated" form is:
$\frac{1-(n+1)x^n+n\cdot x^{n+1}}{(x-1)^2}$ = $1 + 2z + 3x^2 + \cdots + n\cdot x^{n-1}$
>Exam... |
I've been wondering if/when it's possible to "truncate" a series.
>Example 1
For example, the closed form for the series of naturals is:
$\frac{1}{(x-1)^2}$ = $1 + 2z + 3x^2 + \cdots$
The "truncated" form is:
$\frac{1-(n+1)x^n+n\cdot x^{n+1}}{(x-1)^2}$ = $1 + 2z + 3x^2 + \cdots + n\cdot x^{n-1}$
>Exam... |
A variety is usually defined to be the zero set of some polynomials. I don't know much about this, but I think people usually call manifolds that can be realized as varieties "algebraic manifolds". Take for example the polynomial $x^2+y^2-1$, then the zero set of this is $x^2+y^2=1$, or the circle $S^1$. So that is an ... |
I'm trying to compute the following limit:
$\displaystyle \lim_{n \to \infty} \int_{0}^{\infty} (1+\frac{x}{n})^{(-n)} \sin(x/n) dx$
Clearly the integrand converges to 0 pointwise so the trick is to find the dominating function. So for simplicity let $\{f_{n}\}$ denote the sequence of functions of the integrand... |
Here's how I would explain limits:
Think of the function $f(x) = x^{2}$. We might want to ask ourselves: "What is $\lim_{x \rightarrow 3} f(x)$?" Put simply: As $x$ gets near 3, where happens to our function? Clearly, our function goes to 9, which is $3^2$.
One must understand, however, that a limit is a much mor... |
This may be a question whose answer is lost in the mists of time, but why is the elliptical integral of the first kind denoted as $F(\pi/2,m)=K(m)$ when that of the second kind has $E(\pi/2,m)=E(m)$? It's not very consistent! Aside from convention, is there anything stopping us from rationalising these names a little? |
Here's how I would explain limits:
Think of the function $f(x) = x^{2}$. We might want to ask ourselves: "What is $\lim_{x \rightarrow 3} f(x)$?" Put simply: As $x$ gets near 3, where happens to our function? Clearly, our function goes to 9, which is $3^2$.
One must understand, however, that a limit is a much mor... |
I tried to do some math in [a blog post][1] of mine and came to one with a floor function. I wasn't sure how to deal with it so I just ignored it, and then added the ceiling function in my final equation as that seemed to give me the result I wanted. I'm wondering what is the correct way of handling these functions in ... |
How do you handle the floor and ceiling function in an equation? |
Suppose I have two tables, each of unknown size, and I'd like to estimate the average of their true sizes. I hire 2 contractors: one guarantees good precision (i.e., her measurement normally-distributed about the true value, with a standard deviation of 5mm), while the other is a dimwit (i.e., his measurement is also u... |
Im giving a small talk for a combinatorics class on the [Erdos-Szekeres conjecture][1] regarding the happy ending problem (the paper is focused on recent work regarding the conjecture). I always find that when the audience is not up for a long technical proof, and they aren't familiar with the field (in this case comb... |
I''m not entirely sure what your question is. In Conway's
notation **On** denotes the ordinal numbers (and **No** denotes the set of
all surreal Numbers). Basicall the elements of **No** are just von Neumann
ordinals. In von Neumann's construction an ordinal $\alpha$ is just
the set $\\{\beta:\beta<\alpha\\}$ of al... |
I''m not entirely sure what your question is. In Conway's
notation **On** denotes the ordinal numbers (and **No** denotes the set of
all surreal Numbers). Basically the elements of **On** are just von Neumann
ordinals. In von Neumann's construction an ordinal $\alpha$ is just
the set $\\{\beta:\beta<\alpha\\}$ of a... |
I have been convinced for years of the following claim: *Every compact topological space is sequentially compact.* (The converse is trivially false.) I'm positive I have even been taught this in my topology course as an undergraduate, and I can provide at least two wrong, but very subtly so, "proofs".
So this is one... |
Can someone show mathematically how gimbal lock happens when doing matrix rotation with Euler angles for yaw, pitch, roll? I'm having a hard time understanding what is going on even after reading several articles on Google.
Is the only work-around to use quaternions? |
OK, so I got my ships moving along the bezier in time T, now I need to factor in acceleration, but I don't really know how to do it.
I'm using the DeCasteljau Algorithm, that's what my code looks like:
function linp(d, a, b, t) {
d.x = a.x + (b.x - a.x) * t;
d.y = a.y + (b.y - a.y) * t;
... |
On what basis the numbering less than 20, the fraction 17/6 (base 10) does not generate a regular tithe?
|
I have been thinking about the differentials that we use in derivatives and integrals.
For example, I have an equation:
$${\int{w}{dr}} = other stuff$$
The context for this strange equation was:
$q'(t) = a - \frac{b}{r(t)}, r'(t) = c - d r(t) - e q(t)$
$r''(t)= d r'(t) - e q'(t)$
$r''(t)= d r' - e a +... |
OK, so I got my ships moving along the bezier in time T, now I need to factor in acceleration, but I don't really know how to do it.
I'm using the DeCasteljau Algorithm, that's what my code looks like:
function linp(d, a, b, t) {
d.x = a.x + (b.x - a.x) * t;
d.y = a.y + (b.y - a.y) * t;
... |
I am CS major, please be patient if my question is not well-stated.
The dataset is quantitative mass spectrometry data. By labeling proteins of two different samples A and B, we get the relative abundance of proteins in A/B. Alongside with this ratio, we can estimate its variance best on the signal intensities.
W... |
Hypothesis testing on mixture models? |
How do I solve the following question?
> You are given the identity *x² - ax + 144 ≡ (x - b)²*
>
> Work out the values of *a* and *b*. |
Posted by [Pi R Squared][1] on [Yahoo Answers][2]:
> *a = 24* and *b = 12*
>
> *x² - 24x + 144 = (x - 12)²*
[1]: http://in.answers.yahoo.com/my/profile;_ylt=AuGKDZFCaIla44dugPaZqxXaHwx.;_ylv=3?show=31f743411a42c3a8de21b1c9c1b15871aa
[2]: http://in.answers.yahoo.com/question/index?qid=20090510094003AA5Xzgh |
Consider the set S = [0,1/2] U [3/4, 7/8] U [15/16, 31/32] U ...
and the set T = (0,1/2) U (3/4, 7/8) U (15/16, 31/32) U ...
Is S open, closed or neither? Is T open, closed, or neither? |
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